Nowadays an increasing amount of data stored in the public cloud need to be searched remotely for fast accessing. For the sake of privacy, the remote files are usually encrypted, which makes them difficult to be searched by remote servers. It is also harder to efficiently share encrypted data in the cloud than those in plaintext. In this paper, we develop a searchable encryption framework called Linear Encryption with Keyword Search (LEKS) that can semi-generically convert some existing encryption schemes meeting our Linear Encryption Template (LET) to be searchable without re-encrypting all the data. For allowing easy data sharing, we convert a Key-Policy Attributed-Based Encryption (KP-ABE) scheme to a Key-Policy Attributed-Based Keyword Search (KP-ABKS) scheme as a concrete instance of our LEKS framework, making both the encrypted data and the search functionality under fine-grained access control. Notably, the resulting KP-ABKS is the first proven secure ABKS scheme with IND-sCKA security in the random oracle model, assuming the hardness of the ℓ-DCBDH problem derived from the (P, f)-DBDH problem family.